A two-phase free boundary problem for a semilinear elliptic equation

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose Laplacians enjoy a certain inequality‎. ‎We show that in dimension $n=2$‎, ‎solutions have optimal growth at non-isolated singular points‎, ‎and the same result holds for $ngeq3$ under an ($n-1$)-dimensional density condition‎. ‎Furthermore‎, ‎we prove that the set of points in the singular set that the solution does not have optimal growth is locally countably ($n-2$)-rectifiable‎.